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The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them. Some issues are rooted more deeply than others and are discussed in a broader perspective, starting with the function concept. The approach is illustrated using various applications including the transport equation, partial derivatives of a definite integral, and the distortionless (but not lossless) transmission line. This note is suitable for a lecture in any first-year course covering partial derivatives, as a complement to the other course material.

We consider the task of learning a parametric Continuous Time Markov Chain (CTMC) sequence model without examples of sequences, where the training data consists entirely of aggregate steady-state statistics. Making the problem harder, we assume that the states we wish to predict are unobserved in the training data. Specifically, given a parametric model over the transition rates of a CTMC and some known transition rates, we wish to extrapolate its steady state distribution to states that are unobserved. A technical roadblock to learn a CTMC from its steady state has been that the chain rule to compute gradients will not work over the arbitrarily long sequences necessary to reach steady state —from where the aggregate statistics are sampled. To overcome this optimization challenge, we propose ∞-SGD, a principled stochastic gradient descent method that uses randomly-stopped estimators to avoid infinite sums required by the steady state computation, while learning even when only a subset of the CTMC states can be observed. We apply ∞-SGD to a real-world testbed and synthetic experiments showcasing its accuracy, ability to extrapolate the steady state distribution to unobserved states under unobserved conditions (heavy loads, when training under light loads), and succeeding in difficult scenarios where even a tailor-made extension of existing methods fails.

The paper presents a novel approach by using multi- step predictions to address the adaptive sampling problem in a resources and obstacles constrained mobile robotic sensor network to efficiently monitor environmental spatial phenomena. It is first proposed to employ the Gaussian process (GP) to represent the spatial field, which can then be used to predict the field at unmeasured locations. The adaptive sampling problem aims to drive the mobile sensors to optimally navigate the environment where the sensors adaptively take measurements of the spatial phenomena at each sampling step. To this end, a conditional entropy based optimality criterion is proposed, which aims to minimize prediction uncertainty of the GP model. By predicting possible measurements the mobile sensors potentially take in a horizon of multiple sampling steps ahead and exploiting the chain rule of the conditional entropy, a multi-step predictions based adaptive sampling optimization problem is formulated. The objective of the optimization problem is to find the optimal sampling paths for the mobile sensors in multiple sampling steps ahead, which then provides their benefits in terms of better navigation, deployment and data collection with more informative sensor readings. However, the optimization problem is nonconvex, complex, constrained and mixed-integer. Therefore, it is proposed to employ the proximal alternating direction method of multipliers algorithm to efficiently solve the problem. More importantly, the solution obtained by the proposed approach is theoretically guaranteed to be converged to a stationary value. Effectiveness of the proposed algorithm was extensively validated by the real- world dataset, where the obtained results are highly promising.

This paper proposes a compressed hidden naive Bayesian (C-HNB) classifier which is an improved version of hidden naive Bayesian (HNB) by compressing the Bayesian network structure and calculating the attribute correlation with maximal information coefficient (MIC). In C-HNB, we remodel the Bayesian network structure based on the chain rule of joint probability distribution so that the number of hidden parent nodes in C-HNB is smaller than the number of hidden parent nodes in HNB, which reduces the training complexity of Bayesian network. In addition, the attribute correlation in C-HNB is calculated with MIC rather than the mutual information, which makes the Bayesian network more stable because the calculation of mutual information is severely influenced by the discretization of continuous attribute. On the selected KEEL benchmark data sets, we compare the classification performances of C-HNB with HNB. The comparative results show that C-HNB can obtain the better prediction accuracy with the less training time in comparison with HNB and thus the experimental results demonstrate the effectiveness of C-HNB.

Biomarkers play a key role in the monitoring of disease progression. The time taken for an individual to reach a biomarker exceeding or lower than a meaningful threshold is often of interest. Due to the inherent variability of biomarkers, persistence criteria are sometimes included in the definitions of progression, such that only two consecutive measurements above or below the relevant threshold signal that "true" progression has occurred. In previous work, a novel approach was developed, which allowed estimation of the time to threshold using the parameters from a linear mixed model where the residual variance was assumed to be pure measurement error. In this paper, we extend this methodology so that serial correlation can be accommodated. Assuming that the Markov property holds and applying the chain rule of probabilities, we found that the probability of progression at each timepoint can be expressed simply as the product of conditional probabilities. The methodology is applied to a cohort of HIV positive individuals, where the time to reach a CD4 count threshold is estimated. The second application we present is based on a study on abdominal aortic aneurysms, where the time taken for an individual to reach a diameter exceeding 55 mm is studied. We observed that erroneously ignoring the residual correlation when it is strong may result in substantial overestimation of the time to threshold. The estimated probability of the biomarker reaching a threshold of interest, expected time to threshold, and confidence intervals are presented for selected patients in both applications.

A novel approach for parameter estimation in Bayesian networks is presented. The main idea is to introduce a hyper-prior in the Multinomial–Dirichletmodel, traditionally used for conditional distribution estimation in Bayesian networks. The resulting hierarchical model jointly estimates different conditional distributions belonging to the same conditional probability table, thus borrowing statistical strength from each other. An analytical study of the dependence structure a priori induced by the hierarchical model is performed and an ad hoc variational algorithm for fast and accurate inference is derived. The proposed hierarchical model yields a major performance improvement in classification with Bayesian networks compared to traditional models. The proposed variational algorithm reduces by two orders of magnitude the computational time, with the same accuracy in parameter estimation, compared to traditional MCMC methods. Moreover, motivated by a real case study, the hierarchical model is applied to the estimation of Bayesian networks parameters by borrowing strength from related domains.

The use of neural networks in image compression enables transforms and probability models for entropy coding which can process images based on much more complex models than the simple Gauss-Markov models in traditional compression methods. All at the expense of higher computational complexity. In the neural-network based image compression literature, various methods to model the dependencies in the transform domain/latent space are proposed. This work uses an alternative method to exploit the dependencies of the latent representation. The joint density of the latent representation is modeled as a product of conditional densities, which are learned using neural networks. However, each latent variable is not conditioned on all previous latent variables as in the chain rule of factoring joint distributions, but only on a few previous variables, in particular the left, upper and upperleft spatial neighbor variables based on a Markov property assumption for a simpler model and algorthm. The compression performance is comparable with the state- of-the-art compression models, while the conditional densities require a much simpler network and training time due to their simplicity and less number of parameters then its counterparts.

We consider the differential entropy of probability measures absolutely continuous with respect to a given \(\sigma \)-finite “reference” measure on an arbitrary measure space. We state the asymptotic equipartition property in this general case; the result is part of the folklore but our presentation is to some extent novel. Then we study a general framework under which such entropies satisfy a chain rule: disintegrations of measures. We give an asymptotic interpretation for conditional entropies in this case. Finally, we apply our result to Haar measures in canonical relation.

Choosing the stochastic process with the entropy is attributable to increase the least amount of information to the problem under consideration. As a result, the entropy rate for stochastic processes must be determined. In the current correspondence, we will describe the rate of ‘useful’ conditional Renyi information measure, as well as, show that rate for ‘useful’ Renyi Information, the chain rule holds. Therefore, for the rate of ‘useful’ conditional Renyi information and also for an ergodic Markov chain, we will present a relationship and use it to extract the rate of ‘useful’ Renyi information. At last, we will show that Shannon's information rate is the rate for ‘useful’ Renyi information.

When taking care of complex patients with multiple morbidities, accurately predicting the occurrence of each cause-specific event is critical for designing optimal treatment plans. However, standard survival analysis cannot deal with the multiple (usually competing) adverse events and views those competing events as censored. This will result in biased estimation of the incidence rate. In this paper, we propose a deep learning based survival analysis algorithm called DeepComp to jointly predict the progress of the competing events, which can thus inform the doctors which event is more likely to hit the patient first. DeepComp constructs a multi-task recurrent neural network (RNN) and views the conditional probability of each competing event at each time point as the output of each RNN cell. Then the probability chain rule is utilized to combine them together. In this way, the survival probability and the risk for each competing event over the time space are obtained. The multitask structure not only prevents the model from unreasonable censoring but also aids the model in capturing the complex hidden association among the competing events. A novel penalty is added to the loss function to better discriminate the competing risks for each particular patient, which could benefit treatment decision-making. We conduct comprehensive experiments on two real-world clinical data sets and one synthetic data set. The proposed DeepComp method achieves significant performance improvement compared to the state-of-the-art baseline methods.

In this paper, we present a method for convolutional neural network model compression which is based on the removal of filter banks that correspond to unimportant weights. To do this, we depart from the relationship between consecutive layers so as to obtain a factor that can be used to assess the degree upon which each pair of filters are coupled to each other. This allows us to use the unit-response of the coupling between two layers so as to remove pathways int he network that are negligible. Moreover, since the back-propagation gradients tend to diminish as the chain rule is applied from the output to the input layer, here we maximise the variance on the coupling factors while enforcing a monotonicity constraint that assures the most relevant pathways are preserved. We show results on widely used networks employing classification and facial expression recognition datasets. In our experiments, our approach delivers a very competitive trade-off between compression rates and performance as compared to both, the uncompressed models and alternatives elsewhere in the literature. pages = 271–279

Position bias is a critical problem in information retrieval when dealing with implicit yet biased user feedback data. Unbiased ranking methods typically rely on causality models and debias the user feedback through inverse propensity weighting. While practical, these methods still suffer from two major problems. First, when inferring a user click, the impact of the contextual information, such as documents that have been examined, is often ignored. Second, only the position bias is considered but other issues resulted from user browsing behaviors are overlooked. In this paper, we propose an end-to-end Deep Recurrent Survival Ranking (DRSR), a unified framework to jointly model user's various behaviors, to (i) consider the rich contextual information in the ranking list; and (ii) address the hidden issues underlying user behaviors, i.e., to mine observe pattern in queries without any click (non-click queries), and to model tracking logs which cannot truly reflect the user browsing intents (untrusted observation). Specifically, we adopt a recurrent neural network to model the contextual information and estimates the conditional likelihood of user feedback at each position. We then incorporate survival analysis techniques with the probability chain rule to mathematically recover the unbiased joint probability of one user's various behaviors. DRSR can be easily incorporated with both point-wise and pair-wise learning objectives. The extensive experiments over two large-scale industrial datasets demonstrate the significant performance gains of our model comparing with the state-of-the-arts., SIGIR 2020. arXiv admin note: text overlap with arXiv:1809.05818 by other authors

The article is devoted to the memory of Valentin Fedorovich Kolchin. Let ζ, ζi (i ∈ N) be independent identically distributed nonnegative integer-valued random variables, (η i1,…, ηiN ) be the fillings of cells in the generalized scheme of allocation of ζi particles into N cells, 1 ≤ i ≤ n, for fixed Zn = (ζ 1, …, ζn ) these allocation schemes are independent. We consider the conditional probabilities P(A n,N | Zn ) of the event A n,N = {each cell in each of n allocation schemes contains no more than r particles}, where r is some fixed number. The sufficient conditions for the convergence of the sequence P(A n,N | Zn ) to a nonrandom limit with probability 1 are given. It is shown that the random variable ln P(A n,N | Zn ) is asymptotically normal. Applications of the obtained results to the noise-proof encoding are discussed.

In this paper, we propose a soft-output detector for multiple-input multiple-output (MIMO) channels that utilizes achievable information rate (AIR) based partial marginalization (PM). The proposed AIR based PM (AIR-PM) detector has superior performance compared to previously proposed PM designs and other soft-output detectors such as K-best, while at the same time yielding lower computational complexity, a detection latency that is independent of the number of transmit layers, and straightforward inclusion of soft input information. Using a tree representation of the MIMO signal, the key property of the AIRPM is that the connections among all child layers are broken. Therefore, least-square (LS) estimates used for marginalization are obtained independently and in parallel, which have better quality than the zero-forcing decision feedback (ZF-DF) estimates used in previous PM designs. Such a property of the AIRPM detector is designed via a mismatched detection model that maximizes the AIR. Furthermore, we show that the chain rule holds for the AIR calculation, which facilitates an information theoretic characterization of the AIR-PM detector. (Less)

In previous chapters we developed and applied representations for the large deviation analysis of discrete time processes. The derivation of useful representations in this setting follows from a straightforward application of the chain rule. The only significant issue is to decide on the ordering used for the underlying “driving noises” when the chain rule is applied, since controls are allowed to depend on the “past,” which is determined by this ordering.

The chain rule for the classical relative entropy ensures that the relative entropy between probability distributions on multipartite systems can be decomposed into a sum of relative entropies of suitably chosen conditional distributions on the individual systems. Here, we prove a similar chain rule inequality for the quantum relative entropy in terms of channel relative entropies. The new chain rule allows us to solve an open problem in the context of asymptotic quantum channel discrimination: surprisingly, adaptive protocols cannot improve the error rate for asymmetric channel discrimination compared to non-adaptive strategies. In addition, we give examples of quantum channels showing that the channel relative entropy is not additive under the tensor product., Comment: 12 pages

The emergence of real-time auction in online advertising has drawn huge attention of modeling the market competition, i.e., bid landscape forecasting. The problem is formulated as to forecast the probability distribution of market price for each ad auction. With the consideration of the censorship issue which is caused by the second-price auction mechanism, many researchers have devoted their efforts on bid landscape forecasting by incorporating survival analysis from medical research field. However, most existing solutions mainly focus on either counting-based statistics of the segmented sample clusters, or learning a parameterized model based on some heuristic assumptions of distribution forms. Moreover, they neither consider the sequential patterns of the feature over the price space. In order to capture more sophisticated yet flexible patterns at fine-grained level of the data, we propose a Deep Landscape Forecasting (DLF) model which combines deep learning for probability distribution forecasting and survival analysis for censorship handling. Specifically, we utilize a recurrent neural network to flexibly model the conditional winning probability w.r.t. each bid price. Then we conduct the bid landscape forecasting through probability chain rule with strict mathematical derivations. And, in an end-to-end manner, we optimize the model by minimizing two negative likelihood losses with comprehensive motivations. Without any specific assumption for the distribution form of bid landscape, our model shows great advantages over previous works on fitting various sophisticated market price distributions. In the experiments over two large-scale real-world datasets, our model significantly outperforms the state-of-the-art solutions under various metrics., Comment: KDD 2019. The reproducible code and dataset link is https://github.com/rk2900/DLF

This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and Shannon entropies for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article presents an exposit investigation on the information-theoretic and information-geometric characteristics of the new generalized entropy and compare them with the properties of the Tsallis and Shannon entropies.

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-\v{S}mulian theorem are shown., Comment: 29 pages

We present a proof-of-concept of a novel and fully Bayesian methodology designed to detect halos of different masses in cosmological observations subject to noise and systematic uncertainties. Our methodology combines the previously published Bayesian large-scale structure inference algorithm, HADES, and a Bayesian chain rule (the Blackwell-Rao Estimator), which we use to connect the inferred density field to the properties of dark matter halos. To demonstrate the capability of our approach we construct a realistic galaxy mock catalogue emulating the wide-area 6-degree Field Galaxy Survey, which has a median redshift of approximately 0.05. Application of HADES to the catalogue provides us with accurately inferred three-dimensional density fields and corresponding quantification of uncertainties inherent to any cosmological observation. We then use a cosmological simulation to relate the amplitude of the density field to the probability of detecting a halo with mass above a specified threshold. With this information we can sum over the HADES density field realisations to construct maps of detection probabilities and demonstrate the validity of this approach within our mock scenario. We find that the probability of successful of detection of halos in the mock catalogue increases as a function of the signal-to-noise of the local galaxy observations. Our proposed methodology can easily be extended to account for more complex scientific questions and is a promising novel tool to analyse the cosmic large-scale structure in observations., Comment: 17 pages, 13 figures. Accepted for publication in MNRAS following moderate corrections

The aim is to provide a characterization of full conditional measures on a finite Boolean algebra, obtained as lower envelope of the extensions of a full conditional probability defined on another finite Boolean algebra. Such conditional measures are conditional belief functions defined by means of a generalized Bayesian conditioning rule relying on a linearly ordered class of belief functions. This notion of Bayesian conditioning for belief functions is compared with other well-known conditioning rules by looking for those conditional measures that can be seen as lower conditional probabilities.

In this paper, we discuss approaches to the determination of a bivariate distribution by specifying the two conditional density functions, the two conditional survival functions, the two hazard components and the two conditional hazard functions of one variable given the value of the other. In addition, the case in which one considers determination of the joint distribution by specifying the conditional distribution of each variable given that the other variable is bounded below is discussed. Examples are provided in each case.

The problem of two hypothesis testing using Bayesian criterion is considered. In contrast to the standard problem with known conditional distributions of classes and their prior probabilities, the adaptive (or unsupervised) version of the problem is studied where one of the conditional distributions is totally unknown and prior probabilities of hypotheses are unknown also. Observations are available from a mixed distribution only. A decision rule with empirical risk tending to the optimal risk calculated from the complete statistical information is proposed. This decision rule is constructed using methods of kernel non-parametric statistics. The simulation results are given.

The leakage chain rule for quantum min-entropy quantifies the change of min-entropy when one party gets additional leakage about the information source. Herein we provide an interactive version that quantifies the change of min-entropy between two parties, who share an initial classical-quantum state and are allowed to run a two-party protocol. As an application, we prove new versions of lower bounds on the complexity of quantum communication of classical information., A few terminology mistakes were corrected in this version

Plant self‐thinning, which is density‐dependent mortality, has several observed characteristics, including a certain mathematical relationship between growth and density. The original equation that describes self‐thinning is logw¯=C−(3/2)×logdensity, w¯ = mean weight. The basic equation is supported by data from ecology and forestry, but there have been a number of reported slopes that differ from −3/2. This study proposed that change in plant density over time decreases exponentially and that plant growth (weight or volume) increases over time according to one of two models: either exponential growth or sigmoid growth. Exponential growth with a finite time limit, in conjunction with exponential decrease in density, led to the equation log w = C + α/γ × log density, where α/γ

Univariate conditional models are of core importance in supporting medical reasoning, as they allow to decompose a joint probability distribution using the chain rule. Although several methods are available for the elicitation of the joint prior distribution of parameters when the response is a medical categorical variable, the case of a medical continuous response is typically difficult to address, because its sample space is often bounded to an interval and its relationship with explanatory variables may be not linear. In these situations, the elicitation of an informative prior distribution on parameters of a univariate conditional model is challenging, because some level of statistical training is required to a medical expert for interpreting parameters and for retrieving appropriate quantitative information about them. The task can be eased and made efficient by recognizing that physicians typically distinguish among values involving medically normal and pathological patient conditions on the grounds of their personal clinical experience. In this paper, we propose a Generalized Beta regression where parameter elicitation is performed by establishing a correspondence among measured values expressed as relative positions within intervals with a clinical interpretation, regardless the original scales of variables. Software implementing the elicitation procedure is freely available.

A method is proposed for comparing the efficiencies of procedures for sampling the range of values of a random quantity when estimating the probability density. The criterion of efficiency is taken to be an asymptotic expression for the mean square deviation of a regression fit for the probability density.

Summary This paper presents a comparison of three approaches to the teaching of probability to demonstrate how the truth table of elementary mathematical logic can be used to teach the calculations of conditional probabilities. Students are typically introduced to the topic of conditional probabilities—especially the ones that involve Bayes' rule—with the help of such traditional approaches as formula use or conversion to natural frequencies. The truth table approach is an alternative method for explaining the concept and calculation procedure of conditional probability and Bayes' rule.

Using the total probability theorem, we propose a method to calculate the failure rate of a linear vibratory system with random parameters excited by stationary Gaussian processes. The response of such a system is non-stationary because of the randomness of the input parameters. A space-filling design, such as optimal symmetric Latin hypercube sampling or maximin, is first used to sample the input parameter space. For each design point, the output process is stationary and Gaussian. We present two approaches to calculate the corresponding conditional probability of failure. A Kriging metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure and the failure rate of the dynamic system. The proposed method is demonstrated using a vibratory system. Our approach can be easily extended to non-stationary Gaussian input processes.

Extensions of the Kolmogorov convergence criterion and the Marcinkiewicz-Zygmund inequalities from independent random variables to conditional independent ones are derived. As their applications, a conditional version of the Marcinkiewicz-Zygmund strong law of large numbers and a result on convergence in Lp for conditionally independent and conditionally identically distributed random variables are established, respectively.

This study is related to a Condorcetian problem of information aggregation that finds a “true” social ordering using individual orderings, that are supposed to partly contain the “truth”. In this problem, we introduce a new maximum likelihood rule and analyse its performance. This rule selects an alternative that maximizes the probability of realizing individual orderings, conditional on the alternative being the top according to a true social ordering. We show that under a neutrality condition of alternatives, the probability that our rule selects the true top alternative is higher than that of any other rule.

Conditional moment estimates on the cumulative sum of conditionally independent random variables are derived, conditional prophet inequalities for conditionally independent random variables are established, a comparison theorem on conditional moment inequalities between conditionally independent and conditionally negatively associated random variables is established. As applications of these results, a conditional Rosenthal type inequality and two conditional Kolmogorov exponential inequalities for conditionally negatively associated random variables are obtained.

In this study, we propose to use Buckley's confidence interval approach which has not been used before in the literature to calculate marginal and conditional fuzzy probabilities in Bayesian networks. We apply this approach to a real life problem and show that Buckley's confidence interval approach provides to indicate uncertainty better and represents knowledge more explicitly than determining fuzzy probabilities based only on the expert opinion in Bayesian networks.

Logistic regression is a classical linear model for logit-transformed conditional probabilities of a binary target variable. It recovers the true conditional probabilities if the joint distribution of predictors and the target is of log-linear form. Weights-of-evidence is an ordinary logistic regression with parameters equal to the differences of the weights of evidence if all predictor variables are discrete and conditionally independent given the target variable. The hypothesis of conditional independence can be tested in terms of log-linear models. If the assumption of conditional independence is violated, the application of weights-of-evidence does not only corrupt the predicted conditional probabilities, but also their rank transform. Logistic regression models, including the interaction terms, can account for the lack of conditional independence, appropriate interaction terms compensate exactly for violations of conditional independence. Multilayer artificial neural nets may be seen as nested regression-like models, with some sigmoidal activation function. Most often, the logistic function is used as the activation function. If the net topology, i.e., its control, is sufficiently versatile to mimic interaction terms, artificial neural nets are able to account for violations of conditional independence and yield very similar results. Weights-of-evidence cannot reasonably include interaction terms, subsequent modifications of the weights, as often suggested, cannot emulate the effect of interaction terms.

probabilities are introduced as semiring algebraic structures that retain several properties of classical probabilities taking values in the real number interval [0, 1]. Compact probabilities and random variables with such probabilities are mainly studied. Analogs of the Borel-Cantelli lemma and of the law of large numbers are considered. New notions of superposition of probability spaces and superposition of random variables arise on the basis of the Cartesian product of abstract probabilities.

For Markov chains with transition probabilities \(p_{ij}\), the Shannon entropy rate is well-known to be equal to the sum of the \(-\sum _j p_{ij}\log p_{ij}\) weighted by the stationary distribution. This expression derives from the chain rule specific to Shannon entropy. For an ergodic Markov chain, the stationary distribution is the limit of the marginal distributions of the chain.

Accurate kinship predictions using DNA forensic samples has utility for investigative leads, remains identification, identifying relationships between individuals of interest, etc. High throughput sequencing (HTS) of STRs and single nucleotide polymorphisms (SNPs) is enabling the characterization of larger numbers of loci. Large panels of SNP loci have been proposed for improved mixture analysis of forensic samples. While multiple kinship prediction approaches have been established, we present an approach focusing on these large HTS SNP panels for predicting degree of kinship predictions. Formulas for first degree relatives can be multiplied (chained) together to model extended kinship relationships. Predictions are made using these formulations by calculating log likelihood ratios and selecting the maximum likelihood across the possible relationships. With a panel of 30,000 SNPs evaluated on an in silico dataset, this method can resolve parents from siblings and distinguish 1st, 2nd, and 3rd degree relatives from each other and unrelated individuals.

This paper aims to find a suitable decision rule for a binary composite hypothesis-testing problem with a partial or coarse prior distribution. To alleviate the negative impact of the information uncertainty, a constraint is considered that the maximum conditional risk cannot be greater than a predefined value. Therefore, the objective of this paper becomes to find the optimal decision rule to minimize the Bayes risk under the constraint. By applying the Lagrange duality, the constrained optimization problem is transformed to an unconstrained optimization problem. In doing so, the restricted Bayesian decision rule is obtained as a classical Bayesian decision rule corresponding to a modified prior distribution. Based on this transformation, the optimal restricted Bayesian decision rule is analyzed and the corresponding algorithm is developed. Furthermore, the relation between the Bayes risk and the predefined value of the constraint is also discussed. The Bayes risk obtained via the restricted Bayesian decision rule is a strictly decreasing and convex function of the constraint on the maximum conditional risk. Finally, the numerical results including a detection example are presented and agree with the theoretical results.

The Survey of Income and Program Participation (SIPP) is a survey with a longitudinal structure and complex nonignorable design, for which correct estimation requires using the weights. The longitudinal setting also suggests conditional-independence relations between survey variables and early- versus late-wave employment classifications. We state original assumptions justifying an extension of the partially model-based approach of Pfeffermann, Skinner and Humphreys [J. Roy. Statist. Soc. Ser. A 161 (1998) 13–32], accounting for the design of SIPP and similar longitudinal surveys. Our assumptions support the use of log-linear models of longitudinal survey data. We highlight the potential they offer for simultaneous bias-control and reduction of sampling error relative to direct methods when applied to small subdomains and cells. Our assumptions allow us to innovate by showing how to rigorously use only a longitudinal survey to estimate a complex log-linear longitudinal association structure and embed it in cross-sectional totals to construct estimators that can be more efficient than direct estimators for small cells.